Question

The domain of the function $f\left(x\right)=\frac{1}{\sqrt{{\left[x\right]}^2-3\left[x\right]+2}}$ is __________ .

Answer 1

$f\left(x\right)=\frac{1}{\sqrt{{\left[x\right]}^2-3\left[x\right]+2}}$
Since [x]² − 3[x] + 2 = 0
if [x]² − 2[x] − [x] + 2 = 0
[x] ([x] − 2) −1 ([x] − 2) = 0
i.e [x] = 2 or [x] = 1
f(x) is defined only if [x]² − 3[x] + 2 > 0
i.e ([x] −2) ([x]−1) > 0

[x] > 2 or [x] < 1
i.e [x] ≥ 3 or [x] ≤ 0
i.e x ∈[3, ∞) or x∈(−∞, 1)
i.e Domain of f(x) is [3, ∞) ∪ (−∞, 1)
i.e (−∞, 1) ∪ [3, ∞)